Summary: D'Alembertian Series Solutions of LODE with Polynomial Coe cients
S.A. Abramov, Dorodnicyn Computing Centre of RAS, Moscow, Russia
M.A. Barkatou, XLIM UMR CNRS 6172, University of Limoges, France
sabramov@ccas.ru, moulay.barkatou@unilim.fr
Let E be the shift operator acting on sequences of complex numbers as
Ean = an+1 for any sequence (an). The sequence a is d'Alembertian if for
large enough values of the index n the elements an of the sequence satisfy a
linear recurrence equation R(an) = 0, where
R = (E + f1(n)) (E + fk(n)) fi(n) 2 C (n):
Elements of a d'Alembertian sequence can be explicitly represented as a
function of the index n using only rational functions, the gamma function
and nite sums, e. g. the sequence an = 2n n
k=0
(;1)k
;(k+1) is d'Alembertian
with R = (E + 2
n+2) (E ; 2). A d'Alembertian series is a formal power
series 1
n=0 an(z ;z0)n whose coe cients sequence is d'Alembertian (this
notion generalizes the notion of hypergemetric series, where the order k of